The non-analytic momentum dependence of spin susceptibility of Heisenberg magnets in paramagnetic phase and its effect on critical exponents

Abstract

We study momentum dependence of static magnetic susceptibility $\chi(q)$ in paramagnetic phase of Heisenberg magnets and its relation to critical behavior within nonlinear sigma model (NLSM) at arbitrary dimension $2<d<4$. In the first order of $1/N$ expansion, where $N$ is the number of spin components, we find $\chi(q)\propto[q^{2}+\xi^{-2}(1+f(q\xi ))]^{-1+\eta /2}$, where $\xi $ is the correlation length, $q$ is the momentum, measured from magnetic wave vector, the universal scaling function $f(x)$ describes deviation from the standard Landau-Ginzburg momentum dependence. In agreement with previous studies at large $x$ we find $f(x\gg 1)\simeq (2B_{4}/N)x^{4-d}$; the absolute value of the coefficient $B_4$ increases with $d$ at $d>5/2$. Using NLSM, we obtain the contribution of the"anomalous" term $\xi^{-2}f(q\xi )$ to the critical exponent $\nu $, comparing it to the contribution of the non-analytical dependence, originating from the critical exponent $\eta $ (the obtained critical exponents $\nu $ and $\eta $ agree with previous studies). In the range $3\leq d<4$ we find that the former contribution dominates, and fully determines $1/N$ correction to the critical exponent $\nu $ in the limit $d\rightarrow 4.$